I've read about and understand the idea behind the ERC20 Token Standard, but is there an official contract people can download and use? Or is it an official list of functions a smart contract should include in order to be up to standard?

Yes, it is an agreed upon list of functions + events a smart contract must include to meet the standard. See the answer by Eth to this question: ethereum.stackexchange.com/questions/16714/…. In the comments, we asked if there was an official list somewhere. So far, it seems like the Ethereum wiki and the github threads are the only 2 places that list these 6 functions + 2 events that make up the ERC20 standard, and the specific words in which they need to be written. For instance: "TotalSupply" not "TotalCoinSupply"
– TesaMay 29 '17 at 9:16

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The ERC20 standard defines an interface, which is the name of the functions as well as their signature. Signature means a list of their arguments. It also defines what these functions should do. But it does not define how you implement it and thus it does not include any code.

If that's your real name, you have the coolest name ever.
– SentinelMay 4 '17 at 13:52

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"It also defines what these functions should do." < This feels confusing to me, I think what Thorkil meant to say here is that "It also defines what each function returns". So basically it says, if you want to call yourself an ERC-20 you have to offer these functions that return these types of things. Other than that, do what you want!
– Todd VanceJan 22 '18 at 21:28